Electron binding energy of donors in bilayer graphene with a gate-tunable gap

In gapped bilayer graphene, similarly to conventional semiconductors, Coulomb impurities (such as nitrogen donors) may determine the activation energy of its conductivity and provide low-temperature hopping conductivity. However, in spite of the importance of Coulomb impurities, nothing is known about their electron binding energy $E_b$ in the presence of gates. To close this gap, we study numerically the electron binding energy $E_b$ of a singly charged donor in BN-enveloped bilayer graphene with the top and bottom gates at distance $d$ and gate-tunable gap $2\mathrm{\Delta }$. We show that for $10<d<200\mathrm{nm}$ and $1<\mathrm{\Delta }<70\mathrm{meV}$ the ratio $E_b/\mathrm{\Delta }$ changes from 0.4 to 1.4. The ratio $E_b/\mathrm{\Delta }$ stays so close to unity because of the dominating role of the bilayer polarization screening which reduces the Coulomb potential well depth to values $\sim \mathrm{\Delta }$. Still, the ratio $E_b/\mathrm{\Delta }$ somewhat decreases with growing $\mathrm{\Delta }$, faster at small $\mathrm{\Delta }$ and slower at large $\mathrm{\Delta }$. On the other hand, $E_b/\mathrm{\Delta }$ weakly grows with $d$, again faster at small $\mathrm{\Delta }$ and slower at large $\mathrm{\Delta }$. We also studied the effect of trigonal warping and found only a small reduction of $E_b/\mathrm{\Delta }$.

Figure 1

(a) Schematic geometry of dual-gated device studied. hBN substrates are characterized by dielectric constant $\kappa$ and $d$ denotes the distance to gates. (b) The energy dispersion of the two-band model of bilayer graphene without trigonal warping and the definition of binding energy $E_b$ of an in-gap bound state.

Figure 2

The ground-state energy of donor $E$ as a function of $\mathrm{\Delta }$ in units of $\mathrm{\Delta }$ for few values of distance to gates $d$ from 10 to $200\text{nm}$ as well as in the absence of gates.

Figure 3

The ground-state energy of donor $E$ as a function of $\mathrm{\Delta }$ in units of $\mathrm{\Delta }$ for few values of distance to gates $d$ from 10 to $200\text{nm}$ in the absence of screening due to $\sigma$-bond electrons.

Figure 4

(a) A schematic view of the energy spectrum with trigonal warping corrections plotted for gap $\mathrm{\Delta }=5\mathrm{meV}$. Contour lines corresponds to lines of constant energy. Dark contour lines enclose four domains where the minimal energy is realized at $E=\mathrm{\Delta }$. (b) The energy spectrum at $p_y=0$ without (blue solid line) and with trigonal warping corrections (red dashed line).

Figure 5

The donor ground-state energy $E$ in units $\mathrm{\Delta }$ as a function of distance to gates $d$ in the absence (red thick line) and with trigonal warping corrections (blue dashed-dotted line) for $\mathrm{\Delta }=1\text{meV}$ (left panel) and $\mathrm{\Delta }=20\text{meV}$ (right panel). Here the cutoff parameter $n=1$ and the total angular momentum $j=1$. For all values of $d$, the trigonal warping effects result in somewhat higher energy of the electron bound state $E$ and hence smaller donor binding (ionization) energy $E_b=\mathrm{\Delta }-E$.

Figure 6

The average energy (A2) in the two-band model in units ${\gamma }_1$ as a function of distance $x=r/a$ for $\mathrm{\Delta }=1\mathrm{meV}$ ($\delta =0.00256$) and distance $d=13$ nm to gates.